Four books A, B, C, and D are stacked vertically (top to bottom). In how many arrangements are A and B not adjacent to each other?

Introduction / Context:
This is a standard “not adjacent” permutation constraint. We count all arrangements and subtract those with the forbidden adjacency.

Given Data / Assumptions:

• Books are distinct: A, B, C, D.
• Linear (vertical) order matters.
• Adjacency refers to consecutive positions.

Concept / Approach:
Total permutations of 4 distinct books are 4! = 24. Adjacency count is found by treating the adjacent pair (A,B) or (B,A) as a single block and permuting blocks.

Step-by-Step Solution:
All arrangements = 4! = 24.Adjacent arrangements: form a block for A & B. Block permutations: 3! ways to place (AB) with C and D, times 2 internal orders (AB or BA) → 3!*2 = 12.Non-adjacent = 24 − 12 = 12.

Verification / Alternative check:
Direct enumeration or inclusion–exclusion on positions of A and B gives the same result.

Why Other Options Are Wrong:
18, 14, 9 reflect subtracting the wrong adjacency count or forgetting the two internal orders of the (A,B) block.

Common Pitfalls:
Counting AB and BA as one, or forgetting there are only 3! placements of the block with C and D.

Final Answer:
12